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A001254
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Squares of Lucas numbers.
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20
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4, 1, 9, 16, 49, 121, 324, 841, 2209, 5776, 15129, 39601, 103684, 271441, 710649, 1860496, 4870849, 12752041, 33385284, 87403801, 228826129, 599074576, 1568397609, 4106118241, 10749957124, 28143753121, 73681302249, 192900153616, 505019158609, 1322157322201, 3461452808004, 9062201101801, 23725150497409
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 36,60.
Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001. [Note that Identity 34.7 on page 404 is wrong. - Alonso Del Arte, Sep 07 2010]
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
Tanya Khovanova, Recursive Sequences
T. Mansour, A note on sum of k-th power of Horadam's sequence
P. Stanica, Generating functions, weighted and non-weighted sums of powers..., arXiv:math.CO/0010149
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,2,-1)
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FORMULA
| a(n) = (A000032(n))^2.
G.f.: ( 4-7*x-x^2 ) / ( (1+x)*(x^2-3*x+1) ). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 30 2001
a(n)=r^n+(1/r)^n+2*(-1)^n, with r=(3+sqrt(5))/2. a(n+3)=2*a(n+2)+2*a(n+1)-a(n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 08 2003
a(n) = L(2n) + 2(-1)^n = L(n-1)*L(n+1) + 5(-1)^n.
a(n) = 5*Fib(n)^2+4*(-1)^n.
a(n)+a(n+1) = A106729(n). - R. J. Mathar, Nov 17 2011
E.g.f.: 2*exp(-x)*(exp(5*x/2)*cosh(sqrt(5)*x/2)+1). - Wolfdieter Lang, Jan 14 2012
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MAPLE
| with(combinat):seq(5*fibonacci(n)^2+4*(-1)^n, n=0..26)
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MATHEMATICA
| Table[LucasL[n]^2, {n, 0, 29}] (* From Alonso del Arte, Apr 11 2011 *)
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PROG
| (MAGMA) [ Lucas(n)^2 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011
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CROSSREFS
| Cf. A000032, A000204.
Cf. A007598, A079291.
With alternating signs, cf. A075150.
Bisection of A001638 and A006499. First differences of A005970.
Second row of array A103324.
Sequence in context: A069606 A193580 A075150 * A143763 A128626 A193963
Adjacent sequences: A001251 A001252 A001253 * A001255 A001256 A001257
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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